Governor Deval Patrick today proposed a $5 million increase in mental health spending as a means to reduce gun violence, a proposal that springs from a series of mass shootings where the suspects are believed to have been mentally ill, including last month’s massacre in a Connecticut elementary school.

“In the wake of too many tragedies, I have filed legislation to tackle the problem of gun violence and illegal firearm possession,’’ Patrick said in a statement. “Today, we do so again along with an important investment in mental health programs. Mental illness is a disease that can be treated, and our communities are safer when the appropriate services and supports are available for people in need.’’

The proposal would require the state court system to share mental health records from its files with the National Instant Background Check System. Courts deal with mental health issues in both criminal and civil cases.

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In the fiscal 2014 budget that he is expected to unveil in the coming days, Patrick will propose a 3.3 percent increase to the Department of Mental Health budget, which would include the $5 million for programs with the “greatest impact on public safety.’’

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The DMH money would be used for “mobile crisis teams” that would provide mental health services to “individuals in crisis,’’ according to the statement. It would also go toward training for teachers and school administrators — educating them on how to spot mental illnesses in students — and for police and first responders, teaching them how to defuse crises linked to a person with mental illness.

Patrick, in a statement, proposed other changes, including background checks at gun shows, the creation of new criminal penalties in cases wheere guns are used to commit crimes, and limits on purchases of high-powered ammunition.

He will also seek new ways for police and prosecutors to quickly respond — and punish — anyone caught with weapons on school grounds. The plan would create “tiered punishments” based on the type of weapon involved, and would give police the right to arrest without a warrant in order to quickly defuse a dangerous situation at a school.

Patrick’s proposals represent “needed steps toward making our communities safer and supporting those with mental illness,’’ Attorney General Martha Coakley, the state’s top law enforcement official, said in a statement.

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New York Governor Andrew Cuomo on Tuesday signed into law a series of new gun control laws that include provisions to help keep firearms from the mentally ill. New York is the first state to enact changes to its gun laws since the Newtown, Conn., shootings.